The separating variety for the basic representations of the additive group (Q371411)

Assume that an algebraic group \(G\) acts rationally on an irreducible affine variety \(X\) over an algebraically closed field \(k\). This action induces the action on \(k[X]\), the ring of regular functions on \(X\). The algebra of invariants \(k[X]^G\subset k[X]\) consists of the regular functions that are constant on the \(G\)-orbits. An invariant \(f\) is said to separate \(x,y\in X\) if \(f (x) \neq f (y)\). A separating set is a subset of invariants that separates any two elements of \(X\) which are separated by some invariant. The separating variety NEWLINE\[NEWLINES_G = \{(x, y) \in X \times X \,|\, f (x) = f (y) \; \text{for}\;\text{all}\; f \in k[X]^G\} NEWLINE\]NEWLINE describes separating sets as follows. Given the map \(\delta: k[X] \times k[X] \to k[X]\) sending \(f\) to \(f \otimes 1 - 1 \otimes f\), then \(H \subset k[X]^G\) is a separating set if and only if \(V(\delta(H)) = S_G = V(\delta(k[X]^G))\), where \(V\) denotes the common zero set in \(X \times X\) of a set of polynomials. If all the \(G\)-orbits on \(X\) are separated by invariants, then the separating variety is \(\Gamma_G=\{(x,gx)|x\in X, g\in G\}\), the graph of the action of \(G\) on \(X\).NEWLINENEWLINEIn the paper under review it is assumed that \(G\) is the additive group of a field of characteristic zero and \(X\) is the indecomposable rational linear representation of dimension \(n+1\). It is established that the separating variety is the closure of \(\Gamma_G\) and has dimension \(n+2\) unless \(n=2m\) and \(m\geq3\) is odd. In the rest of the cases the separating variety is the union of this closure and another irreducible component of dimension \(n+1\). It is also shown that if the separating variety consists of two irreducible components, then there is no polynomial separating algebra.